Schisma: Difference between revisions

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{{~~interwiki~~

{{Interwiki

| en = 32805/32768

| de = 32805/32768

~~| en = 32805/32768~~

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| Ratio = 32805/32768

| Name = schisma

| Color name = ~~Ly-2~~, ~~Layo comma~~

| Color name = LyM, layoma

| Comma = yes

}}

The '''schisma''', '''32805/32768''', is ~~a small interval about 2 [[cent]]s. It arises as~~ the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])4/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])3/([[64/45]]).

The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])4/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])3/([[64/45]]).

== Other intervals ==

Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]].

~~== History and etymology ==~~

''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio [[887/886]] {{nowrap|(2-1 443-1 887)}}, it is used interchangably with this interval in some of Helmholtz' writing.

== Temperaments ==

~~{{main|Schismatic family}}~~

Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.

== ~~Other intervals~~ ==

=== Nestoria ===

Nestoria tempers out [[361/360]] (S19) and [[513/512]] (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])2 and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]). This corresponds to making 19/16 the minor third and 24/19~19/15 the major third; a good tuning for this is [[65edo]], or if you prefer a more accurate [[19/16]], [[77edo]].

=== Garibaldi ===

Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63|S8]]/[[81/80|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)6 / (2/1). This is derived as the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]).

==== 2.3.5.7.19 subgroup ====

Adding nestoria to garibaldi (tempering [[400/399]] (S20)) results in an extremely elegant temperament which has all of the same patent tunings that garibaldi has but which includes a mapping for 19 through nestoria.

=== 2.3.5.7.17 12 & 118 & 171 (unnamed) ===

As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12 & 118 & 171 temperament:

[[Subgroup]]: 2.3.5.7.17

[[Comma list]]: 1701/1700, 32805/32768

: mapping generators: ~2, ~3, ~7

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg }}

==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====

By tempering [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:

[[Subgroup]]: 2.3.5.7.17.19

[[Comma list]]: 361/360, 513/512, 1701/1700

{{mapping|legend=1| 1 0 15 0 -32 9 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}

: mapping generators: ~2, ~3, ~7

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281

{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}

=== 2.3.5.41 53 & 65 (unnamed) ===

The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = [[1025/1024]] * [[6561/6560]]. Tempering both of these out provides a microtemperament-accuracy mapping for prime 41 via tempering out [[6561/6560|S81]] = (81/80)/(82/81) (the second of the aforementioned commas) s.t any accurate schismic tuning (one with a very slightly flat 81/80) will have a good tuning for an otonal supermajor third [[41/32]] and a flat supermajor second (41/32)/(9/8) = [[41/36]].

~~Commas arising from the difference between~~ a ~~stack~~ of ~~Pythagorean intervals and other primes may also be called schismas~~. The ~~difference between~~ the [[Pythagorean comma]] ~~and~~ [[~~septimal comma~~]] ~~is called~~ the [[~~septimal schisma~~]]. ~~Other examples are~~ [[~~undevicesimal schisma]] and [[Alpharabian schisma~~]].

== History and etymology ==

''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap|(2-1⋅443-1⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing.

== Trivia ==

The schisma explains how the greatly composite numbers 1048576 (220) and 104976 (184) look alike in decimal. The largest common power of two between these numbers is 25, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.

It's also very close in size—about 0.~~0013¢~~ off—from the difference between 3/2 and 7\12, which is about 1.~~9550009¢~~.

It is also very close in size—about 0.0013{{c}} off—from the difference between 3/2 and 7\12, which is about 1.9550009{{c}}. Tempering out this difference instead results in [[atomic]], an extremely high accuracy temperament.

== See also ==

@@ Line 1: / Line 1: @@
-{{interwiki
+{{Interwiki
+| en = 32805/32768
 | de = 32805/32768
-| en = 32805/32768
 | es =
 | ja =
@@ Line 8: / Line 8: @@
 | Ratio = 32805/32768
 | Name = schisma
-| Color name = Ly-2, Layo comma
+| Color name = LyM, layoma
 | Comma = yes
 }}
 {{Wikipedia| Schisma }}
-The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
+The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
+== Other intervals ==
+Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]].
-== History and etymology ==
-''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio [[887/886]] {{nowrap|(2<sup>-1</sup> 443<sup>-1</sup> 887)}}, it is used interchangably with this interval in some of Helmholtz' writing.
 == Temperaments ==
-{{main|Schismatic family}}
 Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.
-== Other intervals ==
+=== Nestoria ===
+{{See also| No-sevens subgroup temperaments #Nestoria }}
+Nestoria tempers out [[361/360]] (S19) and [[513/512]] (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]).  This corresponds to making 19/16 the minor third and 24/19~19/15 the major third; a good tuning for this is [[65edo]], or if you prefer a more accurate [[19/16]], [[77edo]].
+=== Garibaldi ===
+{{Main| Garibaldi }}
+Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63|S8]]/[[81/80|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6</sup> / (2/1). This is derived as the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]).
+==== 2.3.5.7.19 subgroup ====
+{{Main| Garibaldi }}
+Adding nestoria to garibaldi (tempering [[400/399]] (S20)) results in an extremely elegant temperament which has all of the same patent tunings that garibaldi has but which includes a mapping for 19 through nestoria.
+=== 2.3.5.7.17 12 & 118 & 171 (unnamed) ===
+As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12 & 118 & 171 temperament:
+[[Subgroup]]: 2.3.5.7.17
+[[Comma list]]: 1701/1700, 32805/32768
+{{mapping|legend=1| 1 0 15 0 -32 | 0 1 -8 0 21 | 0 0 0 1 1 }}
+: mapping generators: ~2, ~3, ~7
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg }}
+==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====
+By tempering [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:
+[[Subgroup]]: 2.3.5.7.17.19
+[[Comma list]]: 361/360, 513/512, 1701/1700
+{{mapping|legend=1| 1 0 15 0 -32 9 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}
+: mapping generators: ~2, ~3, ~7
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281
+{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}
+=== 2.3.5.41 53 & 65 (unnamed) ===
+The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = [[1025/1024]] * [[6561/6560]]. Tempering both of these out provides a microtemperament-accuracy mapping for prime 41 via tempering out [[6561/6560|S81]] = (81/80)/(82/81) (the second of the aforementioned commas) s.t any accurate schismic tuning (one with a very slightly flat 81/80) will have a good tuning for an otonal supermajor third [[41/32]] and a flat supermajor second (41/32)/(9/8) = [[41/36]].
-Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]] and [[Alpharabian schisma]].
+== History and etymology ==
+''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap|(2<sup>-1</sup>⋅443<sup>-1</sup>⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing.
 == Trivia ==
 The schisma explains how the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike in decimal. The largest common power of two between these numbers is 2<sup>5</sup>, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.
-It's also very close in size—about 0.0013¢ off—from the difference between 3/2 and 7\12, which is about 1.9550009¢.
+It is also very close in size—about 0.0013{{c}} off—from the difference between 3/2 and 7\12, which is about 1.9550009{{c}}. Tempering out this difference instead results in [[atomic]], an extremely high accuracy temperament.
 == See also ==